A criterion for I-adic completeness

Abstract

Let I denote an ideal in a commutative Noetherian ring R. Let M be an R-module. The I-adic completion is defined by $${\hat{M}^I = \varprojlim{}_{\alpha} M/I^{\alpha}M}$$ . Then M is called I-adic complete whenever the natural homomorphism $${M \to \hat{M}^I}$$ is an isomorphism. Let M be I-separated, i.e. $${\cap_{\alpha} I^{\alpha}M = 0}$$ . In the main result of the paper, it is shown that M is I-adic complete if and only if $${{\rm Ext}_R^1(F,M) = 0}$$ for the flat test module $${F = \oplus_{i = 1}^r R_{x_i}}$$ , where $${\{x_1,\ldots,x_r\}}$$ is a system of elements such that $${{\rm Rad} I = {\rm Rad}\, \underline{{\it x}} R}$$ . This result extends several known statements starting with Jensen’s result [9, Proposition 3] that a finitely generated R-module M over a local ring R is complete if and only if $${{\rm Ext}^1_R(F,M) = 0}$$ for any flat R-module F.